A statistical analysis of texture on the COBE-DMR rst year sky maps based on the

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1 Mon. Not. R. Astron. Soc. 000, 1{5 (1995) Genus and spot density in the COBE DMR rst year anisotropy maps S. Torres 1, L. Cayon, E. Martnez-Gonzaez 3 and J.L. Sanz 3 1 Universidad de os Andes and Centro Internaciona de Fisica, Bogota, Coombia. Lawrence Berkeey Laboratory and Center for Partice Astrophysics, Berkeey, CA, USA. 3 Dpto. Fsica Moderna, Universidad de a Cantabria and Instituto Mixto de Fsica de Cantabria, CSIC-U. de Cantabria, Santander (Cantabria), Spain Received ; in origina form ABSTRACT A statistica anaysis of texture on the COBE-DMR rst year sky maps based on the genus and spot number is presented. A generaized statistic is dened in terms of \observabe" quantities: the genus and spot density that woud be measured by dierent cosmic observers. This strategy together with the use of Monte Caro simuations of the temperature uctuations, incuding a the reevant experimenta parameters, represent the main dierence with previous anayses. Based on the genus anaysis we nd a strong anticorreation between the quadrupoe ampitude Q rms P S and the spectra index n of the density uctuation power spectrum at recombination of the form Q rms P S = : 1:7 (4:7 1:3) n K for xed n, consistent with previous works. The resut obtained based on the spot density is consistent with this Q rms P S (n) reation. In addition to the previous resuts we have determined, using Monte Caro simuations, the minimum uncertainty due to cosmic variance for the determination of the spectra index with the genus anaysis. This uncertainty is n 0:. Key words: Cosmic Microwave Background Radiation - Cosmoogy 1 INTRODUCTION The study of texture on Cosmic Microwave Background (CMB) maps can be used to constrain scenarios of gaaxy formation as an aternative technique to the temperature correation anaysis. Severa statistica quantities have been proposed, such as number of spots, contour ength, genus (tota curvature of the iso-temperature contours), and tota area of excursion above a certain eve (Sazhin 1985, Bond & Efstathiou 1987, Vittorio & Juszkiewicz 1987, Coes 1988, Martnez-Gonzaez and Sanz 1989, Gott et a. 1990). At arge anguar scaes (> ) gravitationa potentia uctuations at the cosmic photosphere must eave an imprint on the CMB (Sachs & Wofe 1967), which is manifested as temperature uctuations on the D sphere. There have aready been severa statistica anayses of the COBE data. However, due to the great impact that these data have on cosmoogy and the dicuties of its interpretation caused by the experimenta compexities and the ow signa eve, it is important to ook at these data exhaustivey. We propose and use a new statistic, based on the topoogica characteristics of sky maps, which is directy reated with observabe quantities. Anaysis of the rst year COBE-DMR maps based on the genus have recenty been carried out by Torres (1994a, 1994b) and Smoot et a. (1994). From the rst paper a vaue of n = 1: 0:3 was obtained by tting the coherence ange of the COBE temperature maps for a xed quadrupoe-normaized ampitude of Q rms P S = 16K, assuming a scae-free primordia spectrum of density uctuations P (k) / Q rms P Sk n. Smoot et a. (1994) aow variations of Q and nd a reation between Q and n based aso on genus anaysis. In Smoot et a. (1994) the covariance matrix of the genus at dierent temperature threshods and for severa smoothing scaes is incuded in the minimization. The main dierence of our anaysis with the previous works ies in the direct comparison of the genus and spot density of the COBE data with the vaues obtained in each reaization of a mode with no additiona smoothing of the data. The previous anayses have been performed by comparison between the observed genus and the mean vaues deduced from the cosmoogica modes. Mean vaues of mode parameters are not observabe quantities and therefore are not appropriate to compare with the observed data (we wi see, however, that in some cases a statistic based in the mean vaue of a quantity can be interpreted as the mean vaue of the statistic based on reaizations of that quantity; see section 3.1 beow). In this work we use a dierent statistica anaysis of the genus and spot density data that takes into account the ensembe of reaizations that woud

2 S. Torres, L. Cayon, E. Martnez-Gonzaez and J.L. Sanz be measured by dierent cosmic observers. This approach has been used by Scaramea and Vittorio (1993) with the temperature correation function as the observabe quantity. However, they do not consider a the characteristics of the COBE-DMR experiment such as gaactic cut, pixeization, beam smearing, and smoothing, which is essentia for a precise determination of the vaues of cosmoogica parameters aowed by the data. Improvements in instrument sensitivity are very rapidy reducing experimenta errors, however even in an idea noiseess experiment the measured parametes wi have the uncertainty due to cosmic variance. We have used Monte Caro simuations to study the extent to which cosmic variance obscures the information which can be extracted from the genus anaysis. A simiar consideration for the rms temperature uctuations on dierent anguar scaes resuts in minima uncertainties in the determination of n (White, Krauss & Sik 1993). In section we present how Monte Caro simuations of the COBE-DMR maps are performed. The characteristics of the new statistica method considered in this paper are presented in section 3. The minimum range for the spectra index n, impied by cosmic variance, is obtained in section 4. In section 5 we appy the new method to the genus and spot density of the COBE-DMR data and show the resuts. An interpretation of these resuts in terms of the coherence ange is given in section 6. Finay, a summary of the main concusions is given in section 7. MONTE CARLO SIMULATIONS Simuations that take into account instrument noise, sky coverage, gaactic cut, smearing, pixeization scheme and the DMR beam characteristics were done for a set of cosmoogica modes. Simuated maps of the cosmic signa were generated using an expansion on rea harmonics for the temperature (Smoot et a. 1991) for each one of the 6144 DMR pixes: T (; ) = X = X m=0 k [b ;m cos(m) + b ; m sin(m)] N m W P m (cos ); (1) 1= N m ( + 1)( m)! = 4( + m)! where k = p for m 6= 0 and k = 1 for m = 0; P m (cos ) are the Associated Legendre poynomias; the b ;m coecients are rea stochastic and Gaussian distributed variabes with zero mean and mode dependent variance hb ;mi given by (Abbott & Wise 1984, Bond & Efstathiou 1987) hb ;mi = 4 ( + n 1 ) 9 n ( ) 5 Q rms P S ( + 5 n ) ( 3+n ) : () The weights W` for DMR given by Wright et a. (1994) were used. For each reaization two maps are generated by adding to the cosmic signa the noise corresponding to channes A and B of COBE 53 GHz. The noise is determined by instrument sensitivity and the number of observations per pixe. A sma beam smearing correction is appied in order to take into account the motion of the spacecraft during the 1= second integration time. The two maps are added to form the 1 (A+B) map, Gaussian smoothed (s = :9 ) and nay its genus is cacuated. The agorithm used to compute the spot number density and genus is described esewhere (Torres 1994a; 1994b). 3 THE NEW STATISTICAL METHOD We dene the statistic G associated to the genus and N associated to the number of spots as foows: ( G) k = 5 X 5X i=1 j=1 (G k i G COBE i )M 1 ij (G k j G COBE j ): (3) G k i is the genus for the k-th reaization at threshod eve i, and 5 equay spaced threshod eves were used ( : 3:0; :::; 3:0); G COBE i is the genus for the COBE map at threshod i. It is important to notice that G k i is the genus as measured by a cosmic observer with an instrumenta device ike that aboard COBE and thus the statistic ( G) k so constructed is based on observabe quantities. M ij is the covariance matrix: M ij = 1 N reaiz: X N reaiz: k=1 (G k i hg ii)(g k j hg ji) (4) The M ij matrix was cacuated with the Monte Caro reaizations. A statistic ( N ) k reated with the spot number is dened in a simiar manner as ( G) k by repacing G k i (and simiary G COBE i ) with Ni k (Ni COBE ), the number of spots for the reaization k at threshod i. So each mode is now represented by a distribution of vaues ( G) k and in order to nd the mode that is cosest to the COBE data we have to compare distributions. The simpest way to do this is to use the mean vaue of the distribution < G > (athough other choises ike the mode can be used they are more sensitive to the noise due to the imited number of reaizations ). Notice that considering the mean vaue, < G >, is equivaent to a statistic dened by repacing the genus for each reaization G k i in equation (3) by the expected genus of the mode, except for the constant vaue 5. Another possibiity is to use the Komogorov-Smirnov statistic K to compare the G distributions. That we wi do at the end of section 5 in order to check the resuts. For the moment we wi take the simpest aternative < G > which aso requires ess CPU time to compute. 4 IDEAL EXPERIMENT In this section we consider an idea experiment, that incudes a the COBE-DMR experiment characteristics but assuming a noise-ess radiometer, to eucidate the main properties of the new statistica method presented and to study the cosmic variance associated to the cosmoogica parameters as derived from the genus and spot number. The noise of the COBE-DMR radiometers is taken into account when appying the method to the rea data in next section. The no ergodic character of the CMB temperature random ed on the Cosmic photosphere impies a imitation in the accuracy to determine the statistica properties of the

3 Genus and spot density in the COBE DMR rst year anisotropy maps 3 ed from a singe reaization. Even in the idea case that we were abe to measure the background temperature uctuations in every part of the sphere (no gaactic cut) and with negigibe noise we woud ony be abe to measure the expected vaues of the ed within a certain error bar (cosmic variance). Uncertainties in the two point correation function and in other higher moments due to cosmic variance have aready been cacuated (Scaramea and Vittorio 1990, Cayon, Martnez-Gonzaez and Sanz 1991, White, Krauss and Sik 1993, Srednicki 1993, Gutierrez de a Cruz et a. 1994). We cacuate the cosmic variance uncertainty when anaysing the genus of a hypothetica sky map with the characteristics of the COBE-DMR experiment assuming a noise-ess radiometer and no gaactic contamination. In this situation the genus is independent of the ampitude of the cosmic signa and the ony free parameter for the modes is n. Thus, 7 Monte Caro data sets were generated, one for each vaue of n in the range The estimated vaue of n was obtained using the generaized method described above with each reaization from mode n = 1 taken as the input data (i.e. used as the COBE data in eq. 3).So, from the whoe n = 1 data set we obtain a distribution of n vaues. The dispersion of this distribution gives the uncertainty due to cosmic variance: n 0:. 5 ANALYSIS OF THE COBE-DMR MAPS 5.1 COBE-DMR maps Ony data from the most sensitive radiometers were used (i.e. the DMR 53 GHz). Before any anaysis was done, the reeased maps were processed as foows: 1. The maps were converted to thermodynamic temperature scae (a factor of 1.074).. A monopoe and dipoe function, incuding the sma quadrupoar component of the Dopper eect, was tted to the maps (excuding 30 from the gaactic pane) and subtracted. The dipoe and quadrupoe t is in agreement with Smoot et a. (199), thus providing a check for the integrity of the data and the anaysis software. 3. Finay the sum 1 (A+B) and dierence 1 (A-B) maps were formed and Gaussian smoothed ( s = :9 ) in order to reduce noise. 5. Anaysis To nd the restrictions on Q rms P S and n imposed by the COBE-DMR genus and spot density, a grid of Monte Caro data sets were generated with n in the range 0- (with a step of 0.) and Q rms P S between 5 and 33 K (with a step of K). For each of the simuated 1 (A+B) maps, the statistics G and N were cacuated as indicated in section 3. The number of reaizations was set to 400, which proved to be a safe number after testing for the convergence of the resuts of the reevant quantities. For each mode (i.e. pair of vaues Q rms P S; n) one can buid up, using the Monte Caro data set, the probabiity P G( ^ G) of obtaining a reaization with its G smaer or equa to ^ G. Thus, modes with a high P G( ^ G) for sma ^ G impy that many observers woud measure a texture of their maps very simiar to that of COBE, and the opposite for modes with atter P G( ^ G). The same can be done for the probabiity P N ( ^ N ) based on the spot number. Two Figure 1. Distributions of the G statistic for two modes: n = 0:4, Q rms P S = 19 (soid) and n = 0:8, Q rms P S = 19 (dash). exampes of the distribution of G for modes n = 0:4, Q rms P S = 19K and n = 0:8, Q rms P S = 19K are presented in Fig. 1. For both modes the histogram has a maximum near G 45 and the two distributions are very simiar (as we wi discuss ater, there is not a signicant statistica dierence between them). In order to estabish a simpe criterium to compare probabiities, and so to choose the mode that best t the data, we have used the mean < G > (< N >) vaue of the reaizations for each mode. We then search for the minimum < G > (< N >) in the space of mode parameters (Q rms P S; n). The rst and main resut is the anticorreation found between our estimates of n and Q rms P S which can be approximatey given as a straight ine of the form: Q rms P S = : 1:7 (4:7 1:3) n K for xed n. An expanation of this anticorreation in terms of the coherence ange is given in the next section. To obtain the previous reation between Q rms P S and n, we rst assign an error bar to the vaue of Q rms P S with minimum < G > for xed n by using the generaized method with each reaization from that Q rms P S taken as the input data. We then t a straight ine to the pairs (Q rms P S; n) with minimum < G > considering the corresponding error bars. It is cear from this anaysis that there is a wide range of vaues of n that t the data equay we (as can be seen in gure 1). However it seems that for the < G > (< N >) statistic vaues of n ower than 1 are favored. We have checked the stabiity of the main resut with the number of reaizations by performing an additiona set of 400 simuations for modes which dier from the minimum < G > in 1.5 units (a number bigger than the typica error for a samping of 400 reaizations). It is veried that the (Q rms P S; n) reation is mantained. The error bars for the cosmoogica parameters Q rms P S; n were estimated by using each one of the reaizations for the mode n = 0:4, Q rms P S = 19K (which ies in the ine of degeneracy) as input data repacing COBE's data. The resut is a 68% uncertainty of 0:4 for n and +4 6 for Q rms P S, which we consider as typica error bars for n and Q rms P S. Simiar resuts were found for the number of spots N but for this quantity the generaized anaysis was ess sensitive to variations in the mode parameters.

4 4 S. Torres, L. Cayon, E. Martnez-Gonzaez and J.L. Sanz We now test the robustness of the < > method to discriminate among the various cosmoogica modes by comparing it with a more sophisticated method based on the Komogorov-Smirnov K statistic. The K statistic is used to compare the distribution of vaues ( G) k COBE obtained from equation (3) with the distribution ( G) k constructed by repacing G COBE i in that equation by the genus of the reaization of the same mode, G i, and cacuate the statistic KG for the two distributions. Operating in the same way with a the reaizations we then obtain a distribution of KG vaues for that mode and we use the mean vaue < K G > to compare dierent modes. With the new < K G > minima we are abe to approximatey recover the reation (Q rms P S; n) given above. However we do not nd the trend found with < G > favoring n < 1 modes. Given the reativey poor signa to noise ratio of the DMR rst year data there exists a range of modes which are favored by the data (a the modes which ie in the degeneracy aw) and so a sma improvement in the error bars of < K G > (associated to the imited number of reaizations) to better discriminate among the modes woud require a arge amount of CPU time (we use severa ALPHA DEC 3000). However, with the 4 years COBE data set the signa to noise ratio wi improve reducing the (Q rms P S; n) anticorreation and therefore making the statistic K G more suitabe to discriminate among the fewer remaining modes. Comparison of the K G statistic with the < G > one shows that the former is a much better discriminator among dierent modes. 6 DISCUSSION The reation between Q rms P S and n found in the previous section can be interpreted in terms of the coherence ange of the temperature random ed. The interpretation is based on reations between mean quantities of the ed which do not take into account a the experimenta restrictions and the cosmic variance. Therefore this section shoud be ony considered as an approximate approach to the understanding of the previous resuts. As a function of threshod eve, the mean vaue of the genus for a Gaussian random ed ony depends on the coherence ange of the ed c: hg i = 1= c exp( ) ; c = C(0) C 00 (0) 1= (5) where the threshod eve is given in terms of temperature standard deviations. The coherence ange is dened in terms of the ratio between the correation function and its second derivative at zero ag. If it were possibe to measure CMB anisotropies with idea noise-ess instruments, one coud deduce from the previous expression the intrinsic coherence ange of the underying ed (if no cosmic variance and gaaxy cut were present). The coherence ange of noise, as inferred from the measured genus of the 1 (A-B) map and formua (5) is 4:3 0:1. The coherence ange of the signa and noise 53 1 (A+B) map derived from its genus is 4:9 0:1. In the genera case of a sky map incuding signa and noise c is given by c = P P ( + 1)(C;S + C;N ) ; (6) ( + 1)( + 1)(C;S + C;N ) C = ha i exp( ( + 1) eff ); where ha i = hb ;mi (see eq. ), eff is the eective Gaussian smoothing and C ;S and C ;N are the Legendre coecients for the signa P and noise respectivey (i.e. the coecients in C() = 1 4 ( + 1)CP(cos )). In the extreme case of a noise-ess map ony the spectra form (i.e. n) of the mode contributes to the coherence ange but not the ampitude of the signa. For instance, for the n = 1 mode the DMR radiometers woud measure an eective c 1:4 after beamwidth tering ( 3 ), beam smearing (1:3 ), smoothing ange (:9 ) and pixe size (1, corresponding to the dispersion of a Gaussian with the same area as the pixe of side :6 ) have been taken into account. Beam smearing is the smoothing caused by the motion of the antenna during the 1/ second integration time per measurement. The a ;N noise coecients can be estimated from a east squares t of a harmonic expansion to the 1 (A-B) COBE map. A pure noise sky map woud have a smaer coherence ange, c 4 :, considering a eff given by the smoothing ange, pixe size and beam smearing added in quadrature. This vaue agrees with the one obtained by using equation (5). Moreover, equations () and (6) can be soved numericay to give the theoretica Q rms P S as a function of n and coherence ange, Q theory(n; c). This aw gives an anticorreation between Q and n and reproduces we the empirica reation found in the previous section. Then, one can aso estimate the c corresponding to the 53 1 (A+B) map by tting the Q theory(n; c) to the points on the (Q rms P S; n) pane with minimum < G >. This can be considered the coherence ange of the origina random ed and its vaue is 5:35 0:1. The anticorreation found between Q rms P S and n resuts from the fact that modes with high n tend to give more weight to sma anguar scaes, so in order to keep the same coherence ange for a given instrument noise the ampitude Q rms P S of the spectrum must be decreased as n increases. Notice that by using equation (5) a dierent vaue of the coherence ange was found, as coud be expected due to the uncertainty produced by the cosmic variance and the gaactic cut which were not considered. The correction to this vaue can be cacuated by Monte Caro runs of a random ed with a given c = 5 :35, and then comparing with the one obtained from the genus reation (eq. 5). The correction found is just the dierence between the two estimated vaues of c: the vaue for the random ed of 5 :35 and c estimated by eq. (5) for the COBE data. Finay, we point out that a simiar reasoning woud appy to the resuts obtained from the anaysis based on the number of spots N, simpy by substituting equation (5) by the foowing equation: hn i = c 7 CONCLUSIONS exp( ) erfc(= p ) : (7) Based on a generaized statistic which is dened in terms of the genus (or spot number) that woud be measured by dierent cosmic observers we are abe to constrain the quadrupoe moment Q rms P S and the spectra index n of the density uctuation power spectrum at recombination with the COBE-DMR rst year data. We nd that the two

5 Genus and spot density in the COBE DMR rst year anisotropy maps 5 parameters shoud ie within a region around the straight ine Q rms P S = : 1:7 (4:7 1:3) n for xed n, which corresponds to the ine of constant c = 5 :35. This reation has been obtained with the mean vaue < > and conrmed with the aternative quantity < K > derived from the Komogorov-Smirnov statistic appied to the distributions. This anticorreation is consistent with the resuts of Smoot et a. (1994), using the mean genus of the modes for severa smoothings of the data, and with the one obtained by Sejak and Berstchinger (1993) based on an anaysis of the COBE correation function. We have aso studied the minimum uncertainty due to cosmic variance with which one can obtain the spectra index n when using the genus as the statistica quantity in the comparison with the COBE data. The resut is a 1 error bar of n 0: (for n 1). ACKNOWLEDGEMENTS LC, EMG and JLS acknowegde nantia support from the Human Capita and Mobiity programme of the European Union, contract number ERBCHRXCT90079, and the Spanish DGICYT, project number PB C0-0. ST was supported by the European Union under contract number CI1-CT LC thanks the Fubright Commission for the feowship FU The COBE datasets were deveoped by the NASA Goddard Space Fight Center under the guidance of the COBE Science Working Group and were provided by the NSSDC. REFERENCES Abbott, L. F. & Wise, M. B. 1984, Phys. Lett. B, 135, 79 Bond, J. R. & Efstathiou, G. 1987, MNRAS, 6, 655 Cayon, L., Martnez-Gonzaez & Sanz, J.L. 1991, MNRAS, 53, 599 Coes, P. 1988, MNRAS, 34, 509 Gott, J. R., et a. 1990, ApJ, 35, 1 Gutierrez de a Cruz, C.M., Martnez-Gonzaez, E., Cayon, L., Reboo, R. & Sanz, J.L. 1994, MNRAS, (accepted) Martnez-Gonzaez, E. & Sanz, J. L. 1989, MNRAS, 37, 939 Sachs, R. K. & Wofe, A. N., 1967, ApJ, 147, 73 Sazhin, M. V. 1985, MNRAS, 16, 5p Sejak, U. & Berstchinger, E. 1993, ApJ, 417, L9 Scaramea, R. & Vittorio, N. 1990, ApJ, 353, 37 Scaramea, R. & Vittorio, N. 1993, MNRAS, 63, L17 Smoot, G.F. et a. 1991, ApJ, 371, L1 Smoot, G.F. et a. 199, ApJ, 396, L1 Smoot, G.F. et a. 1994, ApJ (accepted) Srednicki, M. 1993, ApJ, 416, L1 Torres, S. 1994a, ApJ, 43, L9 Torres, S. 1994b, in The CMB Workshop, Edts. N. Mandoesi, et a., Capri 1993, Astrophys. Lett. & Comm. (in press) Vittorio, N. & Juszkiewicz, R. 1987, ApJ, 314, L9 White, M., Krauss, L.M. & Sik, J. 1993, ApJ, 418, 535 Wright, E. L., et a. 1994, ApJ, 40, 1